TY - JOUR

T1 - A numerical proof algorithm for the non-existence of solutions to elliptic boundary value problems

AU - Sekine, Kouta

AU - Nakao, Mitsuhiro T.

AU - Oishi, Shin'ichi

AU - Kashiwagi, Masahide

N1 - Funding Information:
This work was supported by JST CREST Grant Number JPMJCR14D4 . The second author was supported by JSPS KAKENHI Grant Number 18K03434 . We thank the editors and reviewers for providing useful comments that helped to improve the content of this manuscript.
Publisher Copyright:
© 2021 The Author(s)

PY - 2021/11

Y1 - 2021/11

N2 - In 1988, M.T. Nakao developed an algorithm that was based on the fixed-point theorem on Sobolev spaces for the numerical proof of the existence of solutions to elliptic boundary value problems on a bounded domain with a Lipschitz boundary (Nakao (1988) [9]). Thereafter, many researchers reported that the numerical existence proof algorithm to elliptic boundary value problems is actually significant and sufficiently useful. However, the numerical proof of the non-existence of solutions to the problem has hitherto not been considered due to several challenges. The purpose of this paper is to solve these difficulties and to propose an algorithm for the numerical proof of the non-existence of solutions in a closed ball B¯H01(uˆ,ρ)={u∈H01(Ω)|‖u−uˆ‖H01≤ρ} to elliptic boundary value problems. We demonstrate some numerical examples that confirm the usefulness of the proposed algorithm.

AB - In 1988, M.T. Nakao developed an algorithm that was based on the fixed-point theorem on Sobolev spaces for the numerical proof of the existence of solutions to elliptic boundary value problems on a bounded domain with a Lipschitz boundary (Nakao (1988) [9]). Thereafter, many researchers reported that the numerical existence proof algorithm to elliptic boundary value problems is actually significant and sufficiently useful. However, the numerical proof of the non-existence of solutions to the problem has hitherto not been considered due to several challenges. The purpose of this paper is to solve these difficulties and to propose an algorithm for the numerical proof of the non-existence of solutions in a closed ball B¯H01(uˆ,ρ)={u∈H01(Ω)|‖u−uˆ‖H01≤ρ} to elliptic boundary value problems. We demonstrate some numerical examples that confirm the usefulness of the proposed algorithm.

KW - Computer-assisted proof

KW - Elliptic problems

KW - Non-existence proof

KW - Numerical proof

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U2 - 10.1016/j.apnum.2021.06.011

DO - 10.1016/j.apnum.2021.06.011

M3 - Article

AN - SCOPUS:85109210238

VL - 169

SP - 87

EP - 107

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

ER -